Theorem atlatex 17013

Description: Every nonzero element of an atomic lattice is greater than or equal to an atom. (Th. hatomic 11932 analog.)

Hypotheses

Ref	Expression
isatlat.b	|- B = (base` K)
isatlat.l	|- L = (le` K)
isatlat.z	|- Z = (0.` K)
isatlat.a	|- A = (AtomsNEW` K)

Assertion

Ref	Expression
atlatex	|- ((K e. AtLat /\ X e. B /\ X =/= Z) -> E.y e. A yLX)

Distinct variable groups:   y,A   y,K   y,X

Proof of Theorem atlatex

Step	Hyp	Ref	Expression
1		isatlat.b	. . . . 5 |- B = (base` K)
2		isatlat.l	. . . . 5 |- L = (le` K)
3		isatlat.z	. . . . 5 |- Z = (0.` K)
4		isatlat.a	. . . . 5 |- A = (AtomsNEW` K)
5	1, 2, 3, 4	isatlat 17012	. . . 4 |- (K e. AtLat <-> (K e. LatNEW /\ A.x e. B (x =/= Z -> E.y e. A yLx)))
6	5	simprbi 353	. . 3 |- (K e. AtLat -> A.x e. B (x =/= Z -> E.y e. A yLx))
7		neeq1 2024	. . . . 5 |- (x = X -> (x =/= Z <-> X =/= Z))
8		breq2 3342	. . . . . 6 |- (x = X -> (yLx <-> yLX))
9	8	rexbidv 2124	. . . . 5 |- (x = X -> (E.y e. A yLx <-> E.y e. A yLX))
10	7, 9	imbi12d 688	. . . 4 |- (x = X -> ((x =/= Z -> E.y e. A yLx) <-> (X =/= Z -> E.y e. A yLX)))
11	10	rcla4cv 2377	. . 3 |- (A.x e. B (x =/= Z -> E.y e. A yLx) -> (X e. B -> (X =/= Z -> E.y e. A yLX)))
12	6, 11	syl 12	. 2 |- (K e. AtLat -> (X e. B -> (X =/= Z -> E.y e. A yLX)))
13	12	3imp 1061	1 |- ((K e. AtLat /\ X e. B /\ X =/= Z) -> E.y e. A yLX)

Syntax hints:   -> wi 3   /\ w3a 858   = wceq 1298   e. wcel 1300   =/= wne 2017  A.wral 2105  E.wrex 2106   class class class wbr 3338  ` cfv 3998  basecbs 16758  lecple 16759  0.cp0 16832  LatNEWclat 16834  AtomsNEWcatm 16981  AtLatcal 16982

This theorem is referenced by:  atomnle 17016  hl1atom 17040  hlatmstc 17047  cvratlem 17061  cvrat4 17076

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014  df-atlat 16986

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